𝐾-groups generated by 𝐾-spaces

A K K -group G G with identity e e is said to be generated by the K K -space X X if X X is a subspace of G G containing e , X e,X algebraically generates G G , and the canonical morphism from the Graev free K K -group over ( X , e ) (X,e) on-to G G is a quotient morphism. An internal characterization of the topology of such a group G G is obtained, as well as a sufficient condition that a subgroup H H of G G be generated by a subspace Y Y of H H . Several illuminating examples are provided.